In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

Verified step by step guidance

1. <Step 1: Understand the given information.> We are given that \( \sin \theta = \frac{15}{17} \) and \( \theta \) is in quadrant II. In quadrant II, sine is positive, and cosine is negative.
2. <Step 2: Use the Pythagorean identity to find \( \cos \theta \).> The identity is \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{15}{17} \) into the identity: \( \left(\frac{15}{17}\right)^2 + \cos^2 \theta = 1 \). Solve for \( \cos^2 \theta \).
3. <Step 3: Determine \( \cos \theta \).> Since \( \theta \) is in quadrant II, \( \cos \theta \) will be negative. Take the negative square root of the result from Step 2 to find \( \cos \theta \).
4. <Step 4: Use the double angle identity for tangent.> The identity is \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \). First, find \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
5. <Step 5: Substitute \( \tan \theta \) into the double angle identity.> Use the value of \( \tan \theta \) from Step 4 in the identity \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \) to find \( \tan 2\theta \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In this context, the sine of an angle θ is given, which is essential for finding other trigonometric values. Understanding how these functions interrelate is crucial for solving problems involving angles and their measures.

Double Angle Formulas

Double angle formulas are identities that express trigonometric functions of double angles in terms of single angles. For example, the formula for tangent is tan(2θ) = 2tan(θ) / (1 - tan²(θ)). These formulas are vital for calculating the values of trigonometric functions at double angles, especially when the original angle's sine or cosine is known.

Quadrants and Angle Signs

The unit circle is divided into four quadrants, each affecting the signs of the trigonometric functions. In quadrant II, sine is positive while cosine and tangent are negative. Knowing the quadrant in which the angle lies helps determine the signs of the trigonometric values, which is essential for accurately calculating and interpreting the results.